Anisotropic Blistering Instability of Highly Ellipsoidal Shells

Hamid Ebrahimi, Amin Ajdari, Dominic Vella, Arezki Boudaoud, and Ashkan Vaziri

Phys. Rev. Lett. 112, 094302 – Published 6 March 2014

Abstract

The formation of localized periodic structures in the deformation of elastic shells is well documented and is a familiar first stage in the crushing of a spherical shell such as a ping-pong ball. While spherical shells manifest such periodic structures as polygons, we present a new instability that is observed in the indentation of a highly ellipsoidal shell by a horizontal plate. Above a critical indentation depth, the plate loses contact with the shell in a series of well-defined “blisters” along the long axis of the ellipsoid. We characterize the onset of this instability and explain it using scaling arguments, numerical simulations, and experiments. We also characterize the properties of the blistering pattern by showing how the number of blisters and their size depend on both the geometrical properties of the shell and the indentation but not on the shell’s elastic modulus. This blistering instability may be used to determine the thickness of highly ellipsoidal shells simply by squashing them between two plates.

Received 2 September 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.094302

Authors & Affiliations

Hamid Ebrahimi¹, Amin Ajdari², Dominic Vella³, Arezki Boudaoud⁴, and Ashkan Vaziri^1,*

¹Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA
²Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA
³Mathematical Institute, University of Oxford,Woodstock Road, Oxford OX2 6GG, United Kingdom
⁴Laboratoire Reproduction et Développement des Plantes & Laboratoire Joliot-Curie, INRA, CNRS, ENS, Université de Lyon, 46 Allée d’Italie, F-69364 Lyon Cedex 07, France

^*vaziri@coe.neu.edu

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Vol. 112, Iss. 9 — 7 March 2014

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Images

Figure 1
(a) Illustrations of the “ping pong ball instability” in a spherical shell of radius $b$ indented by a rigid plate (gray circle in left panel). Here, a polygonal deformation is observed in simulations (left) and experiments on a plastic spherical cap (right) with dimensionless indentation depth $Z^{'} = Z / t = 13.2$ , $t$ the thickness, and $t / b = 0.01$ . (Adapted from Ref. [1].) (b) Axes of the ellipsoids considered here showing the rotational symmetry about the $x$ axis. (c) Indentation of a highly ellipsoidal shell by a rigid plate causes “anisotropic blistering” of the shell—the contact becomes nonconformal. As the dimensionless indentation $Z^{'} = Z / t$ increases, the pattern develops with additional “blisters” forming. The left panel shows the energy density (high energy density corresponds to regions of compression) as computed in simulations. The right panel shows images from experiments performed on a shell made of polyvinylsiloxane using our mold and wetting technique.
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Figure 2
Regime diagram showing the dependence of the type of deformation observed on the aspect ratio $a / b$ and the indentation depth $Z^{'} = Z / t$ . For $0.5 < a / b ≲ 2$ , mirror buckling is followed by the formation of localized defects [the “ping pong ball instability”, Fig. 1] while for $a / b ≳ 2$ , anisotropic blistering occurs [as shown in Fig. 1]. Points show the boundary between these behaviors determined from simulations for various shell thicknesses (as described in the legend); the dashed line shows a fit for the scaling law Eq. (3), which is valid in the limit of $a / b ≫ 1$ .
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Figure 3
The properties of the blistering pattern. (a) The scaling of blister width as a function of the major axis of the shell (both normalized by the shell thickness) predicted by Eq. (4) is borne out by both numerical simulations (colored points) and experiments (black points). (b) Main figure: the number of blisters and the indentation depth at which that number are first observed approximately reproduce the expected scaling Eq. (5) (shown by the dashed line) both for numerical simulations (colored points) and experiments (black points). Inset: raw data for the number of blisters as a function of indentation depth. In both (a) and (b) different shell ellipticities are encoded by the symbol: $a / b = 3$ (circle), $a / b = 4$ (square), $a / b = 5$ (diamond), $a / b = 6$ (down triangle), $a / b = 8$ (up triangle), and $a / b = 10$ (star). Different shell thicknesses are encoded by the color: $t / b = 0.002$ (red), $t / b = 0.005$ (green), $t / b = 0.01$ (blue), $t / b = 0.02$ (magenta), and $t / b \approx 0.015$ (black—experimental points).
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Figure 4
The indentation force, $P$ , as a function of indentation depth, $Z$ , for $a / b = 5$ , $t / b \approx 0.016$ . The results of numerical simulations with $t / b = 0.016$ , $ν = 0.497$ (solid curve) are shown together with the points at which new blisters form (○). Note that the force is approximately linear for small displacements (prior to anisotropic blistering) but dips sharply at the point of blistering before increasing again. The linear regime is well described by the known result for the indentation of an ellipsoidal shell Eq. (6). Experimental results are shown for shells with $t / b = 0.0134 \pm 0.0025$ ( $△$ ) and $t / b = 0.0165 \pm 0.0025$ ( $□$ ); two runs were performed for each scenario (open and closed symbols) and indicative error bars (resulting from the heterogeneity in shell thickness) are shown.
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